It's a bit more nuanced but you're generally right. Intramuscular vaccines do not provide long term protection from infection of the mucosal tissues of the upper respiratory tract. So they can be infected and shed virus but the lungs themselves get IgG antibodies seeping in from blood serum and protection from serious disease.

If you want to have resident B and T cells in the upper respiratory mucosa generating IgA antibodies and generally killing infected cells right away you need to use an intranasal vaccination. In the UK they have already realized this (and it was known in the 1960s).

https://www.gov.uk/government/publications/long-term-evoluti... page 5, #8.

>"Whilst we feel that current vaccines are excellent for reducing the risk of hospital admission and disease, we propose that research be focused on vaccines that also induce high and durable levels of mucosal immunity in order to reduce infection of and transmission from vaccinated individuals. This could also reduce the possibility of variant selection in vaccinated individuals."

https://science.sciencemag.org/content/373/6553/397 - "the ideal vaccination strategy may use an intramuscular vaccine to elicit a long-lived systemic IgG response and a broad repertoire of central memory B and T cells, followed by an intranasal booster that recruits memory B and T cells to the nasal passages and further guides their differentiation toward mucosal protection, including IgA secretion and tissue-resident memory cells in the respiratory tract."

https://www.nature.com/articles/s41577-021-00550-x "Prospects for durable immune control of SARS-CoV-2 and prevention of reinfection" - any of it.